Flow Meter Tutorial

So you want to measure flow?
The answer would seem to be to
purchase a flowmeter. With fluid
flow defined as the amount of
fluid that travels past a given
location, this would seem to be
straightforward — any flowmeter
would suffice. However, consider
the following equation describing
the flow of a fluid in a pipe.

Q = A x v

Q is flow rate, A is the crosssectional
area of the pipe, and v
is the average fluid velocity in the
pipe. Putting this equation into
action, the flow of a fluid traveling
at an average velocity of a 1 meter
per second through a pipe with a
1 square meter cross-sectional
area is 1 cubic meter per second.
Note that Q is a volume per unit
time, so Q is commonly denoted
as the “volumetric” flow rate.
Now consider the following
equation:

W = rho x Q

Where W is flow rate (again - read
on), and rho is the fluid density.
Putting this equation into action,
the flow rate will be 1 kilogram per
second when 1 cubic meter per
second of a fluid with a density of
1 kilogram per cubic meter is
flowing. (The same can be done
for the commonly-used “pounds”.
Without getting into details — a
pound is assumed to be a mass
unit.) Note that W is a mass per unit
time, so W is commonly denoted as
the “mass” flow rate.
Now — which flow do you want to
measure? Not sure? In some
applications, measuring the
volumetric flow is the thing to do.

Consider filling a tank. Volumetric
flow may be of interest to avoid
overflowing a tank where liquids
of differing densities can be added.
(Then again, a level transmitter
and high level switch/shutoff may
obviate the need for a flowmeter.)
Consider controlling fluid flow into
a process that can only accept a
limited volume per unit time.
Volumetric flow measurement
would seem applicable.

In other processes, mass flow is
important. Consider chemical
reactions where it is desirable to
react substances A, B and C. Of
interest is the number of molecules
present (its mass), not its volume.
Similarly, when buying and selling
products (custody transfer) the
mass is important, not its volume.

PART 2

Having discovered that there are
two types of flow rates (volumetric
and mass), it should not be a
surprise that some flowmeters
measure mass (W) while other
flowmeters measure volume (Q).
However, it is not quite that simple.
Repeating the equations from Part 1
(for convenience), it can be seen
that, assuming A is constant, Q can
be determined by measuring the
average fluid velocity v. Further,
assuming that rho is constant, W
can be determined from Q.

Q = A x v
W = rho x Q

Summarizing, some flowmeters
measure volumetric flow, some
flowmeters measure velocity from
which the volumetric flow is
determined, and some flowmeters
measure mass flow. In addition,
when the density is known or
assumed, mass flow can be
determined from the volumetric
flow, and the volumetric flow can
be determined from the mass flow.
So you just wanted to measure flow
— did you now? It all seemed so
logical and simple at the time. Stick
around — it gets worse.
Some flowmeters use other
principles to infer flow. The most
common of these measurements
measure the velocity head
(1/2 rho v x v) to infer the volumetric
flow. Notice that these flowmeters
do NOT measure volume, do NOT
measure mass, and do NOT
measure velocity — but rather
measure a combination of density
and the square of velocity! Would it
surprise you to discover that this is
a description of (commonly-applied)
head flowmeters, such as orifice
plates, venturis, nozzles...? Further,
in many applications, the inferred
volumetric flow is used to determine
the mass flow. Errors can enter the
measurement process during each
measurement and with each
assumption. Is it any surprise
that plant engineers often have
difficulty closing material balances
in their plants?

Summarizing (again), some
flowmeters measure volume,
some flowmeters measure mass,
some flowmeters measure velocity,
and some flowmeters measure
inferentially. Understand the
difference, but also understand that
careful attention to detail can result
in an inferential measurement that
is better than the others.

PART 3

Volumetric flow is expressed in units
that reflect a volume per unit time.
The example in Part 1 determines
cubic meters and cubic feet per unit
time to be volumetric flow units.
Gallons and liters per unit time are
also volumetric flow units.
Mass flow is expressed in units that
reflect a mass per unit time. The
other example in Part 1 determines
kilograms and pounds per unit time
to be mass flow units. (Without
getting into details — a pound is
assumed to be a mass unit.)
Note that the units of time are
independent of whether volumetric
or mass flow is measured.

Let’s have a quiz.
Are the following volumetric or mass liquid flow units?
gallons per minute
cubic feet per second
liters per minute
kilograms per hour
pounds per hour
grams per minute

Can one have a cubic foot of feathers?
yes/no

Can one have a gallon of feathers?
yes/no

Can one have a kilogram of feathers?
yes/no

If you answered volumetric to the
first three questions, mass to the
next three questions, and yes to
the last three questions, you are
on track.

Consider purchasing fuel for your
car. How does a US gallon of
gasoline purchased on a hot
summer day in Las Vegas, Arizona
compare with a US gallon of
gasoline purchased on a cold winter
night in Anchorage, Alaska? It was
determined that a gallon is a
volumetric unit, so logic would
indicate that the same volume of
gasoline was purchased. Yet the
temperature difference would cause
their densities, and hence their
masses, to be different. Using this
logic, more mass would be obtained
by purchasing gasoline in colder
weather. Thinking locally, one might
conclude that it is more economical
to purchase gasoline during the wee
hours of the morning when the
temperature is coldest.

As you might suspect, such is
not the case. Gasoline pumps
compensate for density variation
that occurs due to temperature,
and in doing so, they measure
the amount (mass) of gasoline
dispensed. Yet, a gallon of cold
gasoline will occupy less volume
than when hot. In essence, the
measurement of a gallon of gasoline
actually refers to its volume at
a given temperature (such as
60 degF). As such, this is really a
mass measurement unit because it
refers to the flow of a specific
substance at a given temperature,
Returning to the quiz, let’s not be so
hasty with the first three questions.
They could be incomplete!

PART 3.1

Part 3 discussed the use of
volumetric units (such as gallons)
to infer mass when the composition
and temperature is known.
The example given was that of
purchasing a gallon of gasoline in a
hot and cold climate. The assertion
was that a gallon of gasoline
purchased in hot and cold climates
might have different sizes due to
their differing temperatures, but
their masses should be the same
because the retail flowmeter is
temperature compensated.

A number of e-mails questioning
this assertion and further
investigation resulted in the
interesting discovery that retail
gasoline flowmeters are not
temperature-compensated in the
United States, but are temperaturecompensated
in Canada. In other
words, either the measured volume
(in the US) or the measured
temperature-corrected volume
(in Canada) is used to infer mass.

Consider the following general
analysis:

1. Air temperature differences
between hot and cold climates
are large. In addition, air
temperature fluctuations between
day and night in a given location
can be large.

2. There is a significant difference
between ground temperatures in
hot and cold climates. However,
ground temperature fluctuations
between day and night in a given
location is very small. Ground
temperature fluctuation between
summer and winter in a given
location is relatively small.

3. Gasoline will be warm when it
leaves the refinery, but will cool
in transport to the retailer’s
underground tank. Given time
in the tank, the temperature of
the gasoline will approach the
ground temperature.

4. Flowmeter calibration is
performed using standard
weights, implying a calibration
to mass.

These statements imply that despite
wide air temperature fluctuations,
the temperature of the gasoline
pumped through the flowmeter
should be nearly the same as the
ground temperature. Because the
ground temperature does not
fluctuate very much, the
temperature variation of the
gasoline will be small throughout
the year, so the mass of a gallon of
gasoline should not vary much
throughout the year from a given
tank. Following this logic, the mass
of a gallon of gasoline sold in
Alaska should be the same as one
sold in Nevada.

Fluctuations in gasoline temperature
cause gasoline density changes.
The magnitude with which these
changes affect measurement
accuracy can be quantified by
performing an uncertainty analysis
to determine if temperature
compensation is appropriate.
An uncertainty analysis for this
measurement would likely reveal a
number of sources of measurement
uncertainty, such as (but not limited
to) the effects of ambient air
temperature, gasoline temperature
leaving the refinery, transport time
from the refinery to the tank, ground
temperature, tank level prior to
filling, the volume of gasoline in the
flowmeter piping, flowmeter piping
temperature, frequency of use,
and composition changes. As a
minimum, such analysis would likely
reveal that the consumer would not
be advised to purchase gasoline
from a tank that was just filled with
warm gasoline. A detailed analysis
may reveal other significant issues.

While this is perhaps more
information than one would like
to know about the subject, this
discussion clearly illustrates the
need to understand the process —
and that the same process may be
different in different locations.
Sometimes ... it’s just not so easy.

PART 3.2

A brief review — Part 3 addressed
mass flow measurement, volumetric
flow measurement, and inferred
mass flow measurement. The
measurement of gasoline was given
as an example of the inferred mass
flow measurement (using volumetric
units). Comments resulted in Part
3.1 that addressed some issues
associated with retail gasoline
measurements. This sparked a
flurry of comments regarding how
gasoline is measured at the pump.
This issue attempts to tie the
comments together, so reading this
issue without having read previous
issues may prove difficult.

Gasoline pumps in the USA
measure volume and are calibrated
using volumetric means. In other
words, they are true volumetric
devices — they measure volume
and indicate gallons. Even the New
York Times offered advice to the
consumer on this one with “... buy
gasoline during the coolest time of
the day — early morning or late
evening — while the gasoline is
most dense...” (New York Times,
September 24, 2001, Empowered II
Smart Energy Management, A clean
car is an efficient car, page 7).

Gasoline pumps in Canada
measure volume. This volume is
then compensated for the actual
temperature to indicate the volume
of the gasoline as if it were a certain
temperature. The compensated
volume is an implied mass
measurement. I suspect (but do
not know) that these pumps are
calibrated using volumetric means
that are temperature compensated.
In other words, they are inferred
mass measurement devices and
are calibrated as such — they
measure volume and indicate in
(temperature-compensated) liters.
In Canada, the inferred mass of the
gasoline received should be the
same (within the limitations of the
equipment) regardless of gasoline
temperature. Note however that
composition differences (and
additives) may cause the density at
a given temperature to be different
than its nominal value. As an
example, a 1% increase in gasoline
density from its nominal value does
not affect the actual volume
measured, but will cause the
inferred mass measurement to be
1% lower than the actual mass flow.

My comments on some readers’
responses follow:

One reader questioned whether
the “wee hours of the am” would be
the time when the gasoline would
be at its lowest temperature in an
underground tank. Thermal lag for
underground gasoline storage
tanks is an issue, but may not be
significant. For science class,
my daughter measured the
temperatures 1 meter above and
1 meter below grade in the
fall/winter (in the New York area).
I seem to remember the ground
temperature changing by only
1-2 degC over a period of months.
The above ground temperature
changed by 20 degC (or more?)
during the same period. This issue
is likely to be significant for above
ground storage tanks (as suggested
by other readers). Note however
that filling the tank may cause
larger (transient) effects caused by
such issues as the quantity and
temperature of the gasoline prior
to fill, and the quantity and
temperature of gasoline added.
Not being able to sell compressed
natural gas measured with a
Coriolis mass flowmeter in kg or
lbm (pounds mass) because it was
not considered ‘marketable’ to the
the public illustrates resistance
to change. By the way, when will
gasoline be sold by the kg or lbm —
or better yet, by the BTU or Joule
(as suggested by another reader)?
I suspect that it will not be soon.

The comments and observations
about beating the measurement
were amusing. Society allows
people to (reasonably) operate in
their own self-interest. Parting with
less money for a product is clearly
in the purchaser’s self-interest.
(Engineers sometimes call this an
“optimization problem”, but that is
an issue for another day.)
Comments on how to beat the
system were inevitable.

The safety point regarding gasoline
expansion causing explosions and
fires (after topping off a gas tank in
a cold climate and then parking in a
warm garage) is important. Virtually
everything is potentially dangerous
— even a small puddle of water that
turns to ice...

Reproduced with permisssion of Spitzer and Boyes, LLC
(www.spitzerandboyes.com)